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Optimisation and experimental verification of startup policies for distillation columns
Computers and Chemical Engineering 28 (2004) 253 /265 www.elsevier.com/locate/compchemeng
Optimisation and experimental verification of startup policies for distillation columns Gu¨nter Wozny, Pu Li * Institute of Process and Plant Technology, Technical University Berlin, KWT9, 10623 Berlin, Germany
Abstract Startup of distillation columns is one of the most difficult operations in the chemical industry. Since the startup often lasts a long period of time, leads to off-spec products, and costs much energy, optimisation of startup operating policies for distillation columns is of great interest. In the last few years, we have carried out both theoretical studies and experimental verifications with the purpose of minimising the startup time of distillation columns. Model-based optimisation as well as real plant implementation is the core of this work. The model is at first validated with experimental data and then used in the optimisation problem formulation. A rigorous column model and a sequential dynamic optimisation approach have been applied to several pilot distillation columns. Experimental results indicate that a significant reduction of the startup time can be achieved by implementing the developed optimal policies. These results demonstrate the applicability of the modelling and optimisation methodology. # 2003 Elsevier Ltd. All rights reserved. Keywords: Startup; Distillation column; Modelling; Optimisation; Experimental verification
1. Introduction Due to its nature of phase transition, large time delay and strong interaction between variables, the startup of distillation columns are one of the most difficult operations in the chemical industry. The startup of industrial columns lasts several to dozens of hours. Since the process is unproductive during the startup period, it is desired to shorten this period by optimising startup operation policies. In spite of its importance, however, very little previous work has been done on startup optimisation for distillation processes, due to the difficulties of combining the theoretical and experimental investigations. Conventionally, the so called direct setting strategy has been used in the process industry: the values of control variables corresponding to the specified steady state are set to the columns for startup and one just waits for the column running to the steady state. A long time period is usually needed for startup by using this strategy. Empirical startup strategies have been proposed for single columns to improve the startup
* Corresponding author. Tel.: /49-30-3142-3418; fax: /49-303142-6915. E-mail address: [email protected] (P. Li). 0098-1354/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0098-1354(03)00181-9
performance. Total reflux (Ruiz, Carmeron & Gani, 1988) and zero reflux (Kruse, Fieg & Wozny, 1996) strategies together with a large reboiler duty have been proposed. The switching time from total or zero reflux to values at the steady state is determined by the criterion proposed by Yasuoka, Nakanisshi and Kunugita (1987), i.e. at the time point when the difference between the temperature at the steady state on some trays and their measured value reaches the minimum. Since a column startup is influenced by many factors such as column structure, the type of trays and packing, component properties in the mixture as well as the top and bottom product specifications, these empirical strategies are suitable only for some specific cases. Therefore, systematic approaches concerning these influential factors are required to solve general startup optimisation problems for distillation columns. This calls for systematic methodologies of modelling and optimisation. The essential difficulty in modelling column startup lies in the fact that it is a quite complicated dynamic process. In most startup models of distillation columns, the three-phase-model proposed by Ruiz et al. (1988) has been used. The procedure of a startup from a cold, empty column to the required operating point consists
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of three phases: (1) heating the column by the rising vapour, (2) filling the trays by the reflux and (3) running the column to the defined steady state. The discontinuous phase is the time period from an empty, cold column to the beginning state of equilibrium. In the second phase, the holdup of the trays is filled from the top to the bottom. Hangos, Hallager, Csaki and Jorgensen (1991) studied the discontinuous phase with a simplified non-equilibrium model. Wang, Li, Wozny and Wang (2001) proposed a model considering the tray-by-tray state transfer from non-equilibrium to equilibrium for the first phase and described the second phase by using two different weirs on the tray. Compared with the first two phases (usually less than 1 h), the third phase requires the longest time and, therefore, possesses the potential to reduce the startup period by developing optimal policies. In addition, hydrodynamic properties on trays and packings in the column play an important role in modelling startup processes and thus should be considered. Although various startup models have been proposed, very few studies have been made to utilise these models for column startup optimisation. Based on an established model, simulation can be made by solving the model equations to study the startup behaviour (Ruiz, Basualdo & Scenns, 1995; Bisowarno & Tade, 2000; Eden, Koggersbol, Hallager & Jorgensen, 2000Wang et al., 2001). For simulation an operating policy during startup has to be defined a priori. This means it may be neither optimal in the sense of minimising the startup time, nor feasible in the sense of holding the process constraints (e.g. the product specifications at the desired steady state). Thus, a mathematical optimisation has to be employed to search for an optimal as well as feasible operating policy. Optimisation approaches to solving large-scale problems have been proposed in the previous studies (Vassiliadis, Pantelides & Sargent, 1994; Cervantis & Biegler, 1998; Li, Arellano-Garcia, Wozny & Reuter, 1998). The basic idea of these approaches is to discretise the dynamic system into a large non-linear programming (NLP) problem so that it can be solved by an NLP solver like sequential quadratic programming (SQP). However, although these dynamic optimisation approaches have been proved efficient in many case studies, to the best of our knowledge, they have not been applied to startup optimisation problems for real distillation columns. Derived from the above review, it can be concluded that the theoretical studies on column startup modelling and optimisation approaches have been well developed. This implies that it is possible to apply these results in real columns for startup optimisation. However, no successful industrial application of these results has been reported. This is because implementation issues have remained a concern, primary due to the mathematical complexity inherent in the modelling and optimisation
approaches. To convince the process industry of the applications, verifications of these theoretical results on real plants have to be made. During the last few years, we have carried out a systematic study including both theoretical and experimental investigations to develop optimal operating policies for distillation columns (Kruse, Fieg & Wozny, 1996; Flender, Fieg & Wozny, 1996; Lo¨we, Li & Wozny, 2000; Wendt, Ko¨nigseder, Li & Wozny, 2002). The aim of the work was to answer two questions: (1) which model is currently suitable for column startup optimisation? (2) Which optimisation approach can be used to solve the startup optimisation problem? Model-based optimisation has been conducted in the work. Three different models are used to describe the startup behaviours of distillation columns. A detailed equilibrium model was validated by experimental studies on the pilot plants and used in the optimisation problem formulation. A sequential NLP approach and the simulated annealing (SA) algorithm were used and modified for optimising column startup. The developed optimal startup policies were verified on different pilot columns. Extensive experiments were conducted to test the optimisation results. As a result, significant reduction of the startup time can be achieved by implementing the optimal operating policy.
2. Modelling column startup In this section we present three different models used for describing column startup operations. A simple twostage model is first considered to estimate the behaviours of startup and to gain a rough insight into the dynamics of distillation columns. As shown in Fig. 1, the model consists only of a total condenser and a reboiler. The dynamic component balance of the system is then: HU
dxB dt
Fxf DxD BxB
(1)
where F , D and B represent the molar flow rate of feed, the distillate and bottom flow, respectively. xf , xD , xB are the composition of these three flows. HU denotes the total molar holdup of the system, while V and L are the vapour as well as liquid flows inside the column. The mass balance and equilibrium of the system can be simply described as F /D/B and xD /KxB (where K
Fig. 1. A two-stage model.
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is the phase equilibrium constant), respectively. To study the trajectory of the reboiler composition xB influenced by the reflux flow L , we assume HU , F , xf , V remain constant during startup. Then the time constant of the composition of the light component in the reboiler xB will be: t
HU F (K 1)(V L)
(2)
It indicates that t will be reduced if the reflux is decreased. Fig. 2 illustrates the reboiler composition profiles caused by the direct setting strategy (setting the steady state value of reflux L /LSP during the whole startup period) and by the zero reflux strategy (setting L /0). These profiles can be received by integrating equation (1) from an initial reboiler composition xB 0 with given value of HU , F , xf , V and the defined reflux flow applied to the representation (2). It is noted that in this model there should be K /1 and V /L . To achieve an optimal startup, a proper switching point from zero reflux to the direct setting is to be found, such that the bottom composition will be along the arrow-pointed trajectory from the initial composition xB 0 to the specified product composition xSP B . The vapour flow V represents the reboiler duty, which has the same impact on xB but in the opposite direction. In the same way, the influence of reflux and reboiler duty on the top composition xD can be analysed. As a result, the optimal operating policy derived from this simple model for distillation column startup is to run the column using a maximal reboiler duty and zero reflux in a period of time and then switch to their steady state value. This is a qualitative result and has been used for real implementation on a pilot packed column and an industrial column for separating a mixture of fatty alcohols. Even using this simple strategy, the startup time can be reduced up to 80% in comparison to the conventional direct setting strategy (Flender, Fieg & Wozny, 1996). The second model we use to describe startup behaviours is a detailed tray-by-tray model composed of dynamic component as well as energy balances, vapour/ liquid equilibrium (VLE) and tray hydraulic relations. Fig. 3 shows a general tray in the column of this model, with the variables xi ,j , yi ,j , Lj , Vj , HUj , Pj , Tj as liquid as well as vapour component composition, liquid as well
Fig. 3. A general tray of the second model.
as vapour flow, holdup, pressure and temperature on the tray. Here i (i /1, . . ., NK ) and j (j /1, . . ., NST ) are the indexes for components and trays, respectively. Qj is the energy received from outside of the tray. It corresponds, respectively, to values of energy loss of the trays, reboiler and condenser duty. Then the model equations for each tray will be: Component balance: d(HUj xi;j ) dt
Lj 1 xi;j 1 Vj yi;j Vj1 yi;j1 Lj xi;j Fj xfi;j
(3)
Phase equilibrium: yi;j hj Ki;j (xi;j ; Tj ; Pj )xi;j (1 hj )yi;j1 Summation equation: NK X
xi;j 1;
i 1
NK X
yi;j 1
(5)
i 1
Energy balance: d(HUj HjL ) dt
L V Lj 1 Hj 1 Vj1 Hj1 Lj HjL Vj HjV
Fj HfL Qj
(6)
Holdup correlation: HUj 8j (xi;j ; Tj ; Lj )
(7)
Pressure drop equation: Pj Pj 1 cj (xi;j 1 ; yi;j ; Tj ; Lj 1 ; Vj )
Fig. 2. Reboiler composition profiles.
(4)
(8)
In addition to the equations (3) /(8), there are auxiliary relations to describe the vapour and liquid L enthalpy (HV j , Hj ), phase equilibrium constant (Ki ,j ), holdup correlation (8j ) and pressure drop correlation (cj ) which are functions of the state variables. Parameters in these correlations can be found in chemical engineering handbooks like Reid, Prausnitz and Poling (1987) and Gmehling, Onken and Arlt (1977). Murphree
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tray efficiency (hj ) is introduced to describe the nonequilibrium behaviour. It is a parameter that can be validated by comparing the simulation results and the operating data. In addition, the heat loss of the column is usually significant and should be considered in the modelling and validated with experimental data. The total model equations formulate a complex large-scale differential algebraic equation (DAE) system. Using this equilibrium model, the startup is described by the time period from the first time point at which the phase equilibrium is reached on all trays to the desired steady state. The starting point corresponds to the state at which the first drop of the liquid of the mixture reaches the top of the column. The third model we proposed is a hybrid model that depicts column startup from a cold, empty state (Wang et al., 2001), which is the extension of the second model. Each tray is described from a non-equilibrium phase in which only mass and energy transfer is taking place to an equilibrium phase in which VLE is reached. The switching point between these two phases is determined by the bubble-point temperature at the operating pressure. Fig. 4 illustrates the state transition of the trays: from the empty cold state (EM)0/liquid accumulation (LA) 0/VLE. It describes the states of trays in the rectifying section during startup. It differs from the stripping section due to the downstream from the feed flow. Using this model the simulation of startup procedures becomes more reliable. For the purpose of optimisation of column startup, a compromise between the model accuracy and the problem solvability has to be made. On the one hand, an accurate model is needed to describe startup behaviours, so that the optimisation results can be so credible that they can be directly implemented on the real plants. On the other hand, a complicated model may lead to a complex optimisation problem, which can not be solved with the existing solution approaches. The second model described above can represent the largest part of the startup period and the available optimisation ap-
Fig. 4. State transition of trays during startup.
proaches can be applied to this model. Therefore, we use the equilibrium model formulated by (3) /(8) to describe distillation columns for startup optimisation in this work.
3. Optimisation approaches In most cases, the aim of optimisation of column startup is to minimise the startup time period. It leads to a dynamic optimisation problem usually with reflux rate and reboiler duty as the decision variables. A general dynamic optimisation problem can be described as: minf (x; u) s:t: g(x; ˙ x; u) 0 h(x; ˙ x; u)]0 x(0) x0 xmin 5x5 xmax umin 5u5umax
(9)
where f, g and h are the objective function, vectors of model equations and process constraints, respectively. x and u are vectors of state and decision variables. Here g includes all equations of the startup model and h represents the process restrictions and predefined steady state specifications. An initial state x0 of the column is required to define the state from which the optimisation problem is considered. Approaches to solve such dynamic optimisation problems usually use a discretisation method to transform the dynamic system into an NLP problem. Collocation on finite elements (Finlayson, 1980) and multiple shooting (Bock & Plitt, 1984) are two common methods for the discretisation. The solution approaches to such problems can be classified into simultaneous approaches (Cervantis & Biegler, 1998), where all discretised variables are included in a huge NLP problem, and sequential approaches (Logsdon & Biegler, 1992; Vassiliadis et al., 1994; Feehery & Barton, 1998), where a simulation step is adopted to compute the dependent variables as well as their gradients and thus only the independents are solved by NLP. Although these approaches have been proved to be able to solve large-scale dynamic optimisation problems, reports on the solution of complex problems such as startup models of distillation columns for separating multi-component mixtures, are rarely found in the literature. We have applied two different approaches to solve the dynamic optimisation problem for startup optimisation. The first one is a gradient-based sequential approach (Li et al., 1998), which is briefly described in the following.
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Using collocation on finite elements the considered time period [t0, tf ] is discretised into time intervals (l /1, . . ., NL ). Collocation on finite elements is used due to its high discretisation accuracy. In each interval the variables on the collocation points (i /1, . . ., NC ) are to be computed. With Lagrangian polynomial approximation the state variables in (9) in interval l will be: NC NC Y NC X X t tj xl (t) xl;i p(t)xl;i (10) i 0 i 0 j 0 ti tj j"i
where ti is the time point corresponding to collocation point i. On the collocation points: xl (ti )
NC X
Pj (ti )xl;j xl;i
i 1; NC
(11)
j 0
x˙ l (ti )
NC X dPj (ti ) j 0
dt
xl;j
i 1; NC
(12)
Principally, the decision variables in (9) can also be approximated with orthogonal polynomials. However, for the ease of practical implementation, we consider the form of piecewise constant controls. After this discretisation, problem (9) is now transformed to (13), which includes state variables on all time intervals and at all collocation points in each interval. Thus it is now a large-scale NLP problem: minf (xl;i ; ul ) s:t:
gl;i (xl;i ; ul ) 0
hl;i (xl;i ; ul )]0 x1;0 x(0) xmin 5xl;i 5xmax umin 5 ul 5umax
(13)
To solve it with a sequential framework, the equality constraints will be eliminated by means of an extra simulation step, i.e. by integrating the model equations with the Newton method. In the integration step, we take advantage of the well-structured sparse Jacobian matrix of the column model in the Gauss-elimination to achieve high computation efficiency. The elimination of the model equations (denoted as xl ,i /8l ,i (ul )) leads to a small optimisation problem described as (14), which includes only the control variables and the inequality constraints. minf (8l;i (ul ); ul ) s:t:
hl;i (8l;i (ul ); ul )]0
umin 5ul 5umax
257
(14)
A two-layer approach to solve this problem is proposed: SQP is used to optimise the independent variables in the optimisation layer, while the discretised state variables are solved in the simulation layer. The independent variables include the controls (which are assumed to have a piecewise constant form in each time interval) as well as the lengths of the time intervals. The analytical gradients of the objective function and the inequality constraints to the controls are computed inside each interval. These gradients will be transferred from an interval to the next interval through the continuity conditions for the state variables. The sparse structure of the related matrices is utilised in the sensitivity computation. For details on the approach we refer to Li et al. (1998). In a recent study (Wendt, Li & Wozny, 2000), a multiple time-scale strategy is proposed to modify this approach for solving strong non-linear problems. The number of time intervals determines the length of each interval and the size of the optimisation problem. The large time intervals should be long enough for the practical realisation as well as for the reduction of the computation time concerning the sensitivity calculation. The small time intervals are adjusted in the simulation layer and their length will be kept more flexible to guarantee the accuracy and convergence in the Newton iteration. This is also important for checking the inequality path constraints between the collocation points inside one time interval, such that the length of the time interval is modified to ensure the constraints to be satisfied at all times. This modification makes it possible for the approach to be applied to strong nonlinear processes such as distillation of mixtures with abnormal VLE behaviours as well as use of column pressure as an optimisation variable. A FORTRAN package DOSOK (dynamic optimisation with SQP and orthogonal collocation) was developed to carry out the computation. This package has been applied to startup optimisation for different distillation columns described in the next section. The second approach we have used is SA, which is a stochastic search method. The advantage of this method is that it does not require sensitivity information and thus can be connected directly to an available simulator (Hanke & Li, 2000; Li, Lo¨we, Arellano-Garcia & Wozny, 2000). Since commercial simulation software is widely used in industry, using SA is an easy way to conduct startup optimisation. The shortcoming of this method is its low computation efficiency, i.e. many runs of simulation are needed to reach the optimal solution. SA is applied to the startup study on a pilot column with 20 bubble-cap trays for separating a methanol-water mixture (see Section 4.2).
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The reason we have used two different approaches to solve the startup optimisation problem is as follows. We have first implemented the startup model into a commercial software being able to carry out dynamic simulation. Thus the SA algorithm could be directly connected to the simulator. The computation time for the startup optimisation of one column (the bubble-cap tray column described in Section 4.2) was several days by a SUN SPARC 10 Station. It would be intolerable to use SA for the startup optimisation of the two-column system (described in Section 4.3). Therefore, we replaced SA with the gradient-based approach. Using this approach, the CPU-time for the startup optimisation of the two-column system was only several hours with the same workstation.
4. Experimental verification on different pilot plants In this section, experimental results on three different pilot plants are presented to verify the modelling and the optimisation approaches for startup of distillation columns. 4.1. A packed column The first pilot column is a packed column (height: 6.5 m, diameter: 0.07 m) which is packed with Sulzer packings (Sulzer Chemtech), 1.5 m BX in the rectifying section and 1.0 m DX in the stripping section. A total condenser is mounted inside the column top. The operating pressure is held with a vacuum pump. A frequency-adjustable divider for discharge (distillate) and return (reflux) of the condensate is employed to manipulate the reflux ratio. The reboiler is an electrical heat exchanger with adjustable heat duty. Several sampling and temperature measuring points are located along the column. The plant is equipped with a process control system. A mixture of two fatty alcohols (1hexanol (C6) and 1-octanol (C8)) is to be separated with the column at a specified steady state point. To startup the column from a cold state (e.g. 25 8C), it is reasonable that the maximum heat duty is to be used to warm up the packing, the column wall and the feed stream until the condensate reaches the top of the column. By using the maximum heat duty (/Q˙/ /1 kW) it takes about 30 min to receive the first drop of the condensate. Since the equilibrium model is used, the task of the optimisation is to determine the policies of the reflux ratio and the reboiler duty after this time point. The conventional direct setting startup procedure used in the chemical industry simply sets the values of the two control variables at the desired steady state (‘‘nominal’’ values). This procedure was tested on the column and it took 7 h to reach the desired steady state. Fig. 5 shows the measured temperature profiles along
Fig. 5. Measured temperature profiles of the packed column by direct setting policy.
the column resulted from this strategy. Kruse, Fieg & Wozny (1996) modified this procedure by first switching the heat duty to the nominal value but without reflux for a period of time, and then switching the reflux ratio to its nominal value. The time point of the second switching is determined by checking the summed minimum quadratic error of the measured and desired temperatures along the column. By implementing this strategy the startup period was reduced to 2 h. The sequential optimisation approach based on the equilibrium model is used to develop optimal startup policies. Corresponding to the height of the packings the number of the theoretical trays is 28 in the rectifying section and 22 in the stripping section. The influence of the packing on the holdup of the internal energy is also added to the model. The model is validated by comparing the data from simulation and experiment. For the dynamic optimisation, 40 time intervals are chosen to discretise the dynamic system. The nominal values corresponding to the steady state (R /1.5, Q˙/ /0.7 kW) are used as the guess profiles of the two control variables. The optimisation results are shown in Figs. 6 and 7 where t /0 corresponds to the time of the first drop of reaching the condenser. During the first 12 min the two
Fig. 6. Computed optimal reflux and reboiler duty policy for the packed column.
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Fig. 7. Computed optimal composition profiles of the packed column.
control variables should be kept at their limiting values (R /0.1, Q˙/ /1 kW) to run the column at the maximum speed. After this period the reflux ratio begins to increase and at the same time the reboiler duty begins to decrease, which slows down the run to prevent an overshooting. By doing so the column is allowed to slide to the desired steady state as fast as possible, because of the improvement of the separation effect. This can be clearly seen from Fig. 7: the distillate composition drops in the first 12 min due to the maximum heating and the minimum reflux, with which the bottom purity is increased quickly. In the second period the distillate composition returns to its specification due to the rising reflux ratio and the decreasing reboiler duty. From the computed results, the total startup period would be 52 min (i.e. 30 min to the first condensate plus 22 min to the steady state). The startup time of the implementation of the optimal policy to the pilot plant was found to be 1 h, as shown in Fig. 8. 4.2. A bubble-cap tray column The second pilot column we used for experimental verification is a tray-by-tray column. The column has a diameter of 100 mm and 20 bubble-cap trays with a
Fig. 8. Measured temperature profiles of the packed column by the optimal policy.
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central downcomer. Isolation coat is mounted to prevent the heat loss from the column wall. The boilup is provided by an electrical reboiler with a maximum duty of 30 kW. The condensation is carried out by a total condenser with cooling water. The plant is equipped with temperature, pressure, level and flow rate measurements and electrical valves for the flow control. All input/output signals are treated by a process control system. Several control loops have been configured and implemented on the plant. The control system is connected to the local area network to manage experimental data. The composition of the feed stream, distillate and bottom product is measured off-line with a gas chromatograph. We consider the separation of a mixture of water and methanol by this plant. The startup of the column to the steady state with a purity of 99.5 mol% for both methanol and water was studied. SA is used for the startup of this plant and the equilibrium model is used in the problem formulation. The model was implemented in the software SPEEDUP as a simulator, which is called by a file of the SA algorithm. The problem definition is the minimisation of the time period from an initial state of the column to the desired steady state. Fig. 9 shows the computed optimal operation policy and Fig. 10 shows the corresponding product purity profiles. It is interesting to note the difference between the optimal policies of the packed column (Fig. 6) and the tray column (Fig. 9). Unlike the operating policy for the packed column shown in Fig. 6, both the reflux rate and reboiler duty shown in Fig. 9 for this tray column should be high in the first period and should be decreased to the steady state nominal value in the second period. Figs. 10 and 11 shows the measured bottom and top temperature profiles by different startup policies: (a) direct setting; (b) zero reflux; (c) optimal policy. All three experiments had the same feed condition (composition: 29 mol% of methanol, flow rate: 15 l/ h, temperature: 60 8C). It can be seen that the time taken for reaching both bottom and top temperature at steady state was 220, 170 and 120 min by the three different policies, respectively, as shown in Fig. 12.
Fig. 9. Computed optimal policy for the bubble tray column.
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Fig. 10. Computed optimal purity profiles for the bubble tray column.
Fig. 13. The heat-integrated column system.
Fig. 11. Measured bottom temperature profiles of the bubble tray column.
of LP. Due to the heat-integration startup of the plant becomes complicated. The end of the startup period is defined as the time when both columns have arrived at the desired steady state. The startup time can be defined as the objective function subject to the steady state conditions. Since this kind of formulation may cause numerical expenses, the following formulation is used as an alternative to describe this optimisation problem: tf
g
2 2 2 LP HP min [(xHP D (xD x ) D (xB x ) B D x ) t0 2 (xLP B ]dt B x )
Fig. 12. Measured top temperature profiles of the bubble tray column.
4.3. A heat-integrated column system As shown in Fig. 13, the third pilot plant considered is a heat-integrated column system consisting of a high pressure (HP) and a low pressure (LP) column, with 28 and 20 bubble-cap trays, respectively. The vapour from HP is introduced as the heating medium to the reboiler
(15)
where xD and xB are the distillate and bottom product specifications, respectively, for the light component. t0 is the initial time point and tf is the final time point. The pilot plant with the parallel arrangement to separate a mixture of methanol and water is considered. The total feed flow is splitted into two parallel flows to the two columns and both columns have top and bottom product. This arrangement represents a typical operation case of such processes in the chemical industry. The input constraints are the limitations of the control variables, i.e. the reflux flow of both HP and LP and the reboiler duty of HP: 0 5LHP (t)5LHP max LP
0 5L (t)5LLP max 0 5QHP (t)5QHP max
(16) (17)
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The feed condition (flow, composition and temperature) is determined through experiment such that it will lead to a proper vapour and liquid load for the two columns. With the equilibrium model, we assume the starting state for the optimisation at the time point when the pressure of HP reaches 2.5 bar, at which both columns just approach VLE and the heat-integration takes place. The state variables at this initial state can be computed with the help of the measured data in the past experiment. Moreover, the profile of the pressure rising in the HP has to be estimated (Wendt et al., 2002). Furthermore, to deal with the time-optimal problem, a sufficiently large time horizon has to be chosen, in order to ensure the plant to reach the steady state. The final optimisation results of the optimal trajectories of the reboiler duty for HP and the reflux flow for both columns are shown in Fig. 14. The corresponding product composition and temperature profiles are shown in Figs. 15 and 16, respectively. As shown in Fig. 14, the optimisation results illustrate that at the beginning a high value of the reboiler duty should be chosen, since it is necessary to increase especially the bottom temperatures and thus the purity of the bottom products. With a slight time delay, the two refluxes need to be increased in order for the two columns to approach the desired purity of both top and bottom products as fast as possible, as shown in Figs. 15 and 16. However, as soon as the column system reaches close to the steady state, the controls need to be step by step decreased down to their steady state level. It should be noted that the optimisation results are developed based on the model. The model reflects the steady state points of the pilot plant fairly well and also depicts roughly the shape of the experimentally measured profiles, but there is still a large model-mismatch concerning the time delay. From the optimisation results, a startup rule for practical purpose can be obtained. It can be seen in Fig. 14 that there is one certain time point at which the control parameters have to be decreased drastically.
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Fig. 15. Computed optimal composition profiles during startup.
Fig. 16. Computed optimal temperature profiles during startup.
When this decrease for the reboiler duty has to be made, the temperature of the bottom in the LP has reached the value of approximately 98 8C. Since this temperature is the major concern for startup operation of this plant, this value can be used as a switching criterion. To transfer the optimisation results to a practical operation policy, an easy-implementing strategy can be derived, which means that the plant is first operated with the maximum value for all control parameters until the bottom of the LP reaches its switching temperature and then all the control parameters are switched to their steady state values.
Fig. 14. Computed optimal trajectories of the control variables.
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For comparison of startup time, different operating policies were implemented in different startup runs for the experimental verification. A feed flow of a methanol /water mixture (F /34.5 l/h, xf /0.3 mol/ mol, Tf /60 8C) was to be separated. The feed flows of the two columns are set as constant with their steady state value (F1 /19 l/h, F2 /15.5 l/h). The feed tray is set at 10th tray for HP and 6th tray for LP. The operating pressure at steady state is 4.7 bar for HP and atmospheric for LP. The startup of the pilot plant from a cold, empty state (at atmospheric temperature and pressure) to the desired steady state (xD /0.99, xB / 0.01 mol/mol for both columns) was considered. Based on simulation as well as experimental results, the bottom and top temperatures of the two columns corresponding to the steady state purity specifications are THP Bss / LP LP 144 8C, THP Tss /108 8C and TBss /99 8C, TTss /65 8C, respectively. And the corresponding values of the control variables at this steady state are QHP ss /9.54 LP kW, LHP ss /26 l/h, Lss /11.2 l/h. During startup, the level and flow control loops shown in Fig. 13 are active, while the temperature control loops (except for the cooling water temperature control) are set to open loop in order to manually implement the startup policies. The direct setting strategy was first implemented for startup. Fig. 17 shows the measured temperature profiles of the two columns. It can be seen that there is a time delay of about 50 min for the top temperature of LP (TLP T ) to begin to increase after the top temperature of HP (THP T ) rises. This is because the driving force of LP is from the latent heat of the vapour from HP. first reached its However, despite of the delay, TLP T steady state value. It took 465 min for all temperatures to approach their steady state. Moreover, it is shown that the warm-up time of the two columns (i.e. the
discontinuous phase) was about 10 and 20 min, respectively. The second strategy studied was the total distillate strategy. In a beginning period the condensed liquid from both columns was pulled out as distillate (zero reflux) and then the operation was switched to the reflux flow for the steady state value. The reboiler duty remained constant during startup. This total distillate strategy has the advantage to accelerate the rise of the bottom temperature of both columns. The switching point from zero reflux to the nominal reflux value was decided by the minimum point of the following function: MT
N X
½Ti Tss;i ½
(19)
i 1
which is the sum of discrepancies of measured temperatures on all trays (there is a temperature sensor on each tray for the pilot plant) and their desired values at the steady state. The value of this function was on-line computed and observed, through which the switching time point can be determined. Fig. 18 shows the measured temperature profiles caused by the total distillate strategy. Due to zero reflux before the switching, the temperature increased with a fast speed. An over-shooting of the top temperature of both columns was observed. Because of the heat-integration, the bottom temperature of LP is stagnated and thus without over-shooting. Switching the reflux flow to their steady state value led to a decrease of the top temperature of both columns. The total startup time for all temperatures to reach their steady state value was reduced to 403 min. The optimal strategy developed by the optimisation was implemented in the third experimental run. From the optimisation results shown in Fig. 14, the control
Fig. 17. Measured temperature profiles by direct setting strategy.
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Fig. 18. Measured temperature profiles by total distillate strategy.
variables (the reflux flow for both columns and the reboiler duty for HP) should be first set at an optimised maximal value for a time period and then decreased gradually to their steady state value. From the practical point of view, it is desired to implement a simple startup policy. Therefore, we simply tailored the numerically optimised operating policy shown in Fig. 14 into a twostage strategy. The optimised maximal values of the HP control variables (QHP opt /11.14 kW, Lopt /29 l/h, LP Lopt /13.5 l/h) were taken in the first period and the steady state values were set in the second period to the pilot plant. The switching time point was chosen at the time when the bottom temperature of LP reaches 98 8C that is almost approaching its steady state value.
Fig. 19 shows the top and bottom temperature profiles of both columns by the optimal strategy. With HP the enhanced values of the control variables, THP B , TT LP and TB rose fast to their steady state value before the switching. After the switching the two columns continued running to the desired steady state. It can be seen that HP ran a little bit over the steady state before the switching in order to provide enough energy to LP, so that both columns could approach the desired steady state as quickly as possible. Compared with the results of the other startup strategies (Figs. 17 and 18), the optimal strategy really resulted in the best startup performance and thus took the shortest startup time. With the optimal strategy, the total startup time was
Fig. 19. Measured temperature profiles by optimal strategy.
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about 300 min, which is 64% of the time needed by the direct setting strategy.
the desired steady state. Moreover, the startup of some distillation columns with special characteristics such as reactive distillation and three-phase distillation is also a future challenge.
5. Conclusions The theoretical work on dynamic process modelling, simulation and optimisation has been well-developed in the last two decades. However, very few studies on the validation and implementation of such results to real plants for column startup have been made in the past. Although there exist profit incentives, no industrial application has been reported on realisation of dynamic optimisation results. Extensive and intensive work on practical verification needs to be conducted to bridge the theoretical development and the industrial application, so as to convince the process industry to apply the theoretical results. To this end, beside the theoretical work of modelling, simulation and optimisation, practical issues including plant and equipment engineering should be considered. In this work, we have considered the startup optimisation problem, which represents one of the most complicated dynamic operations in chemical industry. Dozens of hours are needed in industrial practice for column startup and it results in large amount of off-spec products. Many factors have impacts on the performance of startup operations. There exist no general rules of startup strategies for all kinds of distillation columns. We used a systematic approach including modelling, model validation, optimisation and implementation on the real pants to address this problem. Model-based optimisation for searching time-optimal policies for column startup was the core of this work. A detailed equilibrium model was chosen for the base of optimisation. The model was at first validated (e.g. tray efficiency, tray holdup, column heat loss etc.) and then used for startup policy development. Reflux and reboiler duty policies for startup were searched for and verified on different pilot plants. The temperature profiles on trays of the column were measured as signals to observe the performances by different startup strategies. Significant reduction of startup time was achieved by implementing the optimal policies, in comparison to the conventional startup strategy. The experimental results demonstrated the applicability of the modelling and optimisation results, showing that they can be applied to start up industrial columns. An extension of this work will be the optimisation of column startup based on the non-equilibrium model, so as to include the time period from the cold, empty state to the equilibrium state into the problem formulation. Moreover, heuristics for column startup is to be developed. These heuristics may depend on the properties of equipment (packings, tray-internals), the features of mixtures to be separated, as well as the requirement of
Acknowledgements We thank Deutsche Forschungsgemeinschaft (DFG) for the financial support in this work under the contract WO565/6-3 and WO565/10-3.
References Bisowarno, B. H., & Tade, M. O. (2000). Dynamic simulation of startup in ethyl tert-butyl ether reactive distillation with input multiplicity. Industrial and Engineering Chemistry Research 39 , 1950. Bock, H. G., & Plitt, J. (1984). A multiple shooting algorithm for direct solution of optimal control problems, IFAC 9th world congress . Budapest, Hungary, 2 /6 July 1984. Cervantis, A., & Biegler, L. T. (1998). Large-scale DAE optimization using a simultaneous NLP formulation. American Institute of Chemical Engineering Journal 44 , 1038. Eden, M. R., Koggersbol, A., Hallager, L., & Jorgensen, S. B. (2000). Dynamics and control during startup of heat integrated distillation column. Computers and Chemical Engineering 24 , 1091. Feehery, Y. W. F., & Barton, P. I. (1998). Dynamic optimization with state variable path constraints. Computers and Chemical Engineering 22 , 1241. Finlayson, B. A. (1980). Nonlinear analysis in chemical engineering . New York: McGraw-Hill. Flender, M., Fieg, G., & Wozny, G. (1996). Classification of new product changeover strategy (NPS) for different application of distillation columns. Computers and Chemical Engineering 20 (Suppl), S1131. Gmehling, J., Onken, U., & Arlt, W. (1977). VLE data collection . Frankfurt: DECHEMA. Hangos, K.M., Hallager, L., Csaki, Z.S., & Jorgensen, S.B. (1991). A qualitative model for simulation of the startup of a distillation column with energy feedback. In: L. Puigjaner & A. Espuna (Eds.), Computer-oriented process engineering , pp. 87 /92. Elsevier. Hanke, M., & Li, P. (2000). Simulated annealing for the optimization of batch distillation processes. Computers and Chemical Engineering 24 , 1. Kruse, C., Fieg, G., & Wozny, G. (1996). A new time-optimal strategy for column startup and product changeover. Journal of Process Control 6 , 187. Li, P., Arellano-Garcia, H., Wozny, G., & Reuter, E. (1998). Optimization of a semibatch distillation process with model validation on the industrial site. Industrial and Engineering Chemistry Research 37 , 1341. Li, P., Lo¨we, K., Arellano-Garcia, H., & Wozny, G. (2000). Integration of simulated annealing to a simulation tool for dynamic optimization of chemical processes. Chemical Engineering Processing 39 , 357. Logsdon, J. S., & Biegler, L. T. (1992). Decomposition strategies for large-scale dynamic optimization problems. Chemical Engineering Science 47 , 851. Lo¨we, K., Li, P., & Wozny, G. (2000). Development and experimental verification of a time-optimal startup strategy for a high purity distillation column. Chemical Engineering Technology 23 , 841.
G. Wozny, P. Li / Computers and Chemical Engineering 28 (2004) 253 /265 Reid, R. C., Prausnitz, J. M., & Poling, B. E. (1987). The properties of gases and liquids . New York: McGraw-Hill. Ruiz, A., Basualdo, M. S., & Scenns, N. J. (1995). Reactive distillation dynamic simulation. Industrial and Engineering Chemistry Research 73 (A), 363. Ruiz, A., Carmeron, I., & Gani, R. (1988). A generalized dynamic model for distillation columns-III. Study of startup operations. Computers and Chemical Engineering 12 , 1. Vassiliadis, V. S., Pantelides, C. C., & Sargent, R. W. H. (1994). Solution of a class of multistage dynamic optimization problems, 1. Problems without path constraints. Industrial Engineering and Chemical Research 33 , 2111. Wang, L., Li, P., Wozny, G., & Wang, S. Q. (2001). Simulation of startup operation for batch batch distillation starting from a cold
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state, American Institute of Chemical Engineering Annual Meeting , 4 /9 November 2001, Reno, Nevada, paper 85h. Wendt, M., Ko¨nigseder, R., Li, P., & Wozny, G. (2002). Theoretical and experimental studies on startup strategies for a heatintegrated distillation column system, International conference on distillation and absorption . Baden-Baden, 30.0902.10.2002. Wendt, M., Li, P., & Wozny, G. (2000). Batch distillation optimization with a multiple time-scale sequential approach for strong nonlinear processes. In: S. Pierucci, Proceeding of ESCAPE-10 , ComputerAided Chemical Engineering 8 , 121. Yasuoka, H., Nakanisshi, E., & Kunugita, E. (1987). Design of an online startup system for a distillation column based on a simple algorithm. International Chemical Engineering 27 , 466.
Optimisation and experimental verification of startup policies for distillation columns Gu¨nter Wozny, Pu Li * Institute of Process and Plant Technology, Technical University Berlin, KWT9, 10623 Berlin, Germany
Abstract Startup of distillation columns is one of the most difficult operations in the chemical industry. Since the startup often lasts a long period of time, leads to off-spec products, and costs much energy, optimisation of startup operating policies for distillation columns is of great interest. In the last few years, we have carried out both theoretical studies and experimental verifications with the purpose of minimising the startup time of distillation columns. Model-based optimisation as well as real plant implementation is the core of this work. The model is at first validated with experimental data and then used in the optimisation problem formulation. A rigorous column model and a sequential dynamic optimisation approach have been applied to several pilot distillation columns. Experimental results indicate that a significant reduction of the startup time can be achieved by implementing the developed optimal policies. These results demonstrate the applicability of the modelling and optimisation methodology. # 2003 Elsevier Ltd. All rights reserved. Keywords: Startup; Distillation column; Modelling; Optimisation; Experimental verification
1. Introduction Due to its nature of phase transition, large time delay and strong interaction between variables, the startup of distillation columns are one of the most difficult operations in the chemical industry. The startup of industrial columns lasts several to dozens of hours. Since the process is unproductive during the startup period, it is desired to shorten this period by optimising startup operation policies. In spite of its importance, however, very little previous work has been done on startup optimisation for distillation processes, due to the difficulties of combining the theoretical and experimental investigations. Conventionally, the so called direct setting strategy has been used in the process industry: the values of control variables corresponding to the specified steady state are set to the columns for startup and one just waits for the column running to the steady state. A long time period is usually needed for startup by using this strategy. Empirical startup strategies have been proposed for single columns to improve the startup
* Corresponding author. Tel.: /49-30-3142-3418; fax: /49-303142-6915. E-mail address: [email protected] (P. Li). 0098-1354/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0098-1354(03)00181-9
performance. Total reflux (Ruiz, Carmeron & Gani, 1988) and zero reflux (Kruse, Fieg & Wozny, 1996) strategies together with a large reboiler duty have been proposed. The switching time from total or zero reflux to values at the steady state is determined by the criterion proposed by Yasuoka, Nakanisshi and Kunugita (1987), i.e. at the time point when the difference between the temperature at the steady state on some trays and their measured value reaches the minimum. Since a column startup is influenced by many factors such as column structure, the type of trays and packing, component properties in the mixture as well as the top and bottom product specifications, these empirical strategies are suitable only for some specific cases. Therefore, systematic approaches concerning these influential factors are required to solve general startup optimisation problems for distillation columns. This calls for systematic methodologies of modelling and optimisation. The essential difficulty in modelling column startup lies in the fact that it is a quite complicated dynamic process. In most startup models of distillation columns, the three-phase-model proposed by Ruiz et al. (1988) has been used. The procedure of a startup from a cold, empty column to the required operating point consists
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of three phases: (1) heating the column by the rising vapour, (2) filling the trays by the reflux and (3) running the column to the defined steady state. The discontinuous phase is the time period from an empty, cold column to the beginning state of equilibrium. In the second phase, the holdup of the trays is filled from the top to the bottom. Hangos, Hallager, Csaki and Jorgensen (1991) studied the discontinuous phase with a simplified non-equilibrium model. Wang, Li, Wozny and Wang (2001) proposed a model considering the tray-by-tray state transfer from non-equilibrium to equilibrium for the first phase and described the second phase by using two different weirs on the tray. Compared with the first two phases (usually less than 1 h), the third phase requires the longest time and, therefore, possesses the potential to reduce the startup period by developing optimal policies. In addition, hydrodynamic properties on trays and packings in the column play an important role in modelling startup processes and thus should be considered. Although various startup models have been proposed, very few studies have been made to utilise these models for column startup optimisation. Based on an established model, simulation can be made by solving the model equations to study the startup behaviour (Ruiz, Basualdo & Scenns, 1995; Bisowarno & Tade, 2000; Eden, Koggersbol, Hallager & Jorgensen, 2000Wang et al., 2001). For simulation an operating policy during startup has to be defined a priori. This means it may be neither optimal in the sense of minimising the startup time, nor feasible in the sense of holding the process constraints (e.g. the product specifications at the desired steady state). Thus, a mathematical optimisation has to be employed to search for an optimal as well as feasible operating policy. Optimisation approaches to solving large-scale problems have been proposed in the previous studies (Vassiliadis, Pantelides & Sargent, 1994; Cervantis & Biegler, 1998; Li, Arellano-Garcia, Wozny & Reuter, 1998). The basic idea of these approaches is to discretise the dynamic system into a large non-linear programming (NLP) problem so that it can be solved by an NLP solver like sequential quadratic programming (SQP). However, although these dynamic optimisation approaches have been proved efficient in many case studies, to the best of our knowledge, they have not been applied to startup optimisation problems for real distillation columns. Derived from the above review, it can be concluded that the theoretical studies on column startup modelling and optimisation approaches have been well developed. This implies that it is possible to apply these results in real columns for startup optimisation. However, no successful industrial application of these results has been reported. This is because implementation issues have remained a concern, primary due to the mathematical complexity inherent in the modelling and optimisation
approaches. To convince the process industry of the applications, verifications of these theoretical results on real plants have to be made. During the last few years, we have carried out a systematic study including both theoretical and experimental investigations to develop optimal operating policies for distillation columns (Kruse, Fieg & Wozny, 1996; Flender, Fieg & Wozny, 1996; Lo¨we, Li & Wozny, 2000; Wendt, Ko¨nigseder, Li & Wozny, 2002). The aim of the work was to answer two questions: (1) which model is currently suitable for column startup optimisation? (2) Which optimisation approach can be used to solve the startup optimisation problem? Model-based optimisation has been conducted in the work. Three different models are used to describe the startup behaviours of distillation columns. A detailed equilibrium model was validated by experimental studies on the pilot plants and used in the optimisation problem formulation. A sequential NLP approach and the simulated annealing (SA) algorithm were used and modified for optimising column startup. The developed optimal startup policies were verified on different pilot columns. Extensive experiments were conducted to test the optimisation results. As a result, significant reduction of the startup time can be achieved by implementing the optimal operating policy.
2. Modelling column startup In this section we present three different models used for describing column startup operations. A simple twostage model is first considered to estimate the behaviours of startup and to gain a rough insight into the dynamics of distillation columns. As shown in Fig. 1, the model consists only of a total condenser and a reboiler. The dynamic component balance of the system is then: HU
dxB dt
Fxf DxD BxB
(1)
where F , D and B represent the molar flow rate of feed, the distillate and bottom flow, respectively. xf , xD , xB are the composition of these three flows. HU denotes the total molar holdup of the system, while V and L are the vapour as well as liquid flows inside the column. The mass balance and equilibrium of the system can be simply described as F /D/B and xD /KxB (where K
Fig. 1. A two-stage model.
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is the phase equilibrium constant), respectively. To study the trajectory of the reboiler composition xB influenced by the reflux flow L , we assume HU , F , xf , V remain constant during startup. Then the time constant of the composition of the light component in the reboiler xB will be: t
HU F (K 1)(V L)
(2)
It indicates that t will be reduced if the reflux is decreased. Fig. 2 illustrates the reboiler composition profiles caused by the direct setting strategy (setting the steady state value of reflux L /LSP during the whole startup period) and by the zero reflux strategy (setting L /0). These profiles can be received by integrating equation (1) from an initial reboiler composition xB 0 with given value of HU , F , xf , V and the defined reflux flow applied to the representation (2). It is noted that in this model there should be K /1 and V /L . To achieve an optimal startup, a proper switching point from zero reflux to the direct setting is to be found, such that the bottom composition will be along the arrow-pointed trajectory from the initial composition xB 0 to the specified product composition xSP B . The vapour flow V represents the reboiler duty, which has the same impact on xB but in the opposite direction. In the same way, the influence of reflux and reboiler duty on the top composition xD can be analysed. As a result, the optimal operating policy derived from this simple model for distillation column startup is to run the column using a maximal reboiler duty and zero reflux in a period of time and then switch to their steady state value. This is a qualitative result and has been used for real implementation on a pilot packed column and an industrial column for separating a mixture of fatty alcohols. Even using this simple strategy, the startup time can be reduced up to 80% in comparison to the conventional direct setting strategy (Flender, Fieg & Wozny, 1996). The second model we use to describe startup behaviours is a detailed tray-by-tray model composed of dynamic component as well as energy balances, vapour/ liquid equilibrium (VLE) and tray hydraulic relations. Fig. 3 shows a general tray in the column of this model, with the variables xi ,j , yi ,j , Lj , Vj , HUj , Pj , Tj as liquid as well as vapour component composition, liquid as well
Fig. 3. A general tray of the second model.
as vapour flow, holdup, pressure and temperature on the tray. Here i (i /1, . . ., NK ) and j (j /1, . . ., NST ) are the indexes for components and trays, respectively. Qj is the energy received from outside of the tray. It corresponds, respectively, to values of energy loss of the trays, reboiler and condenser duty. Then the model equations for each tray will be: Component balance: d(HUj xi;j ) dt
Lj 1 xi;j 1 Vj yi;j Vj1 yi;j1 Lj xi;j Fj xfi;j
(3)
Phase equilibrium: yi;j hj Ki;j (xi;j ; Tj ; Pj )xi;j (1 hj )yi;j1 Summation equation: NK X
xi;j 1;
i 1
NK X
yi;j 1
(5)
i 1
Energy balance: d(HUj HjL ) dt
L V Lj 1 Hj 1 Vj1 Hj1 Lj HjL Vj HjV
Fj HfL Qj
(6)
Holdup correlation: HUj 8j (xi;j ; Tj ; Lj )
(7)
Pressure drop equation: Pj Pj 1 cj (xi;j 1 ; yi;j ; Tj ; Lj 1 ; Vj )
Fig. 2. Reboiler composition profiles.
(4)
(8)
In addition to the equations (3) /(8), there are auxiliary relations to describe the vapour and liquid L enthalpy (HV j , Hj ), phase equilibrium constant (Ki ,j ), holdup correlation (8j ) and pressure drop correlation (cj ) which are functions of the state variables. Parameters in these correlations can be found in chemical engineering handbooks like Reid, Prausnitz and Poling (1987) and Gmehling, Onken and Arlt (1977). Murphree
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tray efficiency (hj ) is introduced to describe the nonequilibrium behaviour. It is a parameter that can be validated by comparing the simulation results and the operating data. In addition, the heat loss of the column is usually significant and should be considered in the modelling and validated with experimental data. The total model equations formulate a complex large-scale differential algebraic equation (DAE) system. Using this equilibrium model, the startup is described by the time period from the first time point at which the phase equilibrium is reached on all trays to the desired steady state. The starting point corresponds to the state at which the first drop of the liquid of the mixture reaches the top of the column. The third model we proposed is a hybrid model that depicts column startup from a cold, empty state (Wang et al., 2001), which is the extension of the second model. Each tray is described from a non-equilibrium phase in which only mass and energy transfer is taking place to an equilibrium phase in which VLE is reached. The switching point between these two phases is determined by the bubble-point temperature at the operating pressure. Fig. 4 illustrates the state transition of the trays: from the empty cold state (EM)0/liquid accumulation (LA) 0/VLE. It describes the states of trays in the rectifying section during startup. It differs from the stripping section due to the downstream from the feed flow. Using this model the simulation of startup procedures becomes more reliable. For the purpose of optimisation of column startup, a compromise between the model accuracy and the problem solvability has to be made. On the one hand, an accurate model is needed to describe startup behaviours, so that the optimisation results can be so credible that they can be directly implemented on the real plants. On the other hand, a complicated model may lead to a complex optimisation problem, which can not be solved with the existing solution approaches. The second model described above can represent the largest part of the startup period and the available optimisation ap-
Fig. 4. State transition of trays during startup.
proaches can be applied to this model. Therefore, we use the equilibrium model formulated by (3) /(8) to describe distillation columns for startup optimisation in this work.
3. Optimisation approaches In most cases, the aim of optimisation of column startup is to minimise the startup time period. It leads to a dynamic optimisation problem usually with reflux rate and reboiler duty as the decision variables. A general dynamic optimisation problem can be described as: minf (x; u) s:t: g(x; ˙ x; u) 0 h(x; ˙ x; u)]0 x(0) x0 xmin 5x5 xmax umin 5u5umax
(9)
where f, g and h are the objective function, vectors of model equations and process constraints, respectively. x and u are vectors of state and decision variables. Here g includes all equations of the startup model and h represents the process restrictions and predefined steady state specifications. An initial state x0 of the column is required to define the state from which the optimisation problem is considered. Approaches to solve such dynamic optimisation problems usually use a discretisation method to transform the dynamic system into an NLP problem. Collocation on finite elements (Finlayson, 1980) and multiple shooting (Bock & Plitt, 1984) are two common methods for the discretisation. The solution approaches to such problems can be classified into simultaneous approaches (Cervantis & Biegler, 1998), where all discretised variables are included in a huge NLP problem, and sequential approaches (Logsdon & Biegler, 1992; Vassiliadis et al., 1994; Feehery & Barton, 1998), where a simulation step is adopted to compute the dependent variables as well as their gradients and thus only the independents are solved by NLP. Although these approaches have been proved to be able to solve large-scale dynamic optimisation problems, reports on the solution of complex problems such as startup models of distillation columns for separating multi-component mixtures, are rarely found in the literature. We have applied two different approaches to solve the dynamic optimisation problem for startup optimisation. The first one is a gradient-based sequential approach (Li et al., 1998), which is briefly described in the following.
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Using collocation on finite elements the considered time period [t0, tf ] is discretised into time intervals (l /1, . . ., NL ). Collocation on finite elements is used due to its high discretisation accuracy. In each interval the variables on the collocation points (i /1, . . ., NC ) are to be computed. With Lagrangian polynomial approximation the state variables in (9) in interval l will be: NC NC Y NC X X t tj xl (t) xl;i p(t)xl;i (10) i 0 i 0 j 0 ti tj j"i
where ti is the time point corresponding to collocation point i. On the collocation points: xl (ti )
NC X
Pj (ti )xl;j xl;i
i 1; NC
(11)
j 0
x˙ l (ti )
NC X dPj (ti ) j 0
dt
xl;j
i 1; NC
(12)
Principally, the decision variables in (9) can also be approximated with orthogonal polynomials. However, for the ease of practical implementation, we consider the form of piecewise constant controls. After this discretisation, problem (9) is now transformed to (13), which includes state variables on all time intervals and at all collocation points in each interval. Thus it is now a large-scale NLP problem: minf (xl;i ; ul ) s:t:
gl;i (xl;i ; ul ) 0
hl;i (xl;i ; ul )]0 x1;0 x(0) xmin 5xl;i 5xmax umin 5 ul 5umax
(13)
To solve it with a sequential framework, the equality constraints will be eliminated by means of an extra simulation step, i.e. by integrating the model equations with the Newton method. In the integration step, we take advantage of the well-structured sparse Jacobian matrix of the column model in the Gauss-elimination to achieve high computation efficiency. The elimination of the model equations (denoted as xl ,i /8l ,i (ul )) leads to a small optimisation problem described as (14), which includes only the control variables and the inequality constraints. minf (8l;i (ul ); ul ) s:t:
hl;i (8l;i (ul ); ul )]0
umin 5ul 5umax
257
(14)
A two-layer approach to solve this problem is proposed: SQP is used to optimise the independent variables in the optimisation layer, while the discretised state variables are solved in the simulation layer. The independent variables include the controls (which are assumed to have a piecewise constant form in each time interval) as well as the lengths of the time intervals. The analytical gradients of the objective function and the inequality constraints to the controls are computed inside each interval. These gradients will be transferred from an interval to the next interval through the continuity conditions for the state variables. The sparse structure of the related matrices is utilised in the sensitivity computation. For details on the approach we refer to Li et al. (1998). In a recent study (Wendt, Li & Wozny, 2000), a multiple time-scale strategy is proposed to modify this approach for solving strong non-linear problems. The number of time intervals determines the length of each interval and the size of the optimisation problem. The large time intervals should be long enough for the practical realisation as well as for the reduction of the computation time concerning the sensitivity calculation. The small time intervals are adjusted in the simulation layer and their length will be kept more flexible to guarantee the accuracy and convergence in the Newton iteration. This is also important for checking the inequality path constraints between the collocation points inside one time interval, such that the length of the time interval is modified to ensure the constraints to be satisfied at all times. This modification makes it possible for the approach to be applied to strong nonlinear processes such as distillation of mixtures with abnormal VLE behaviours as well as use of column pressure as an optimisation variable. A FORTRAN package DOSOK (dynamic optimisation with SQP and orthogonal collocation) was developed to carry out the computation. This package has been applied to startup optimisation for different distillation columns described in the next section. The second approach we have used is SA, which is a stochastic search method. The advantage of this method is that it does not require sensitivity information and thus can be connected directly to an available simulator (Hanke & Li, 2000; Li, Lo¨we, Arellano-Garcia & Wozny, 2000). Since commercial simulation software is widely used in industry, using SA is an easy way to conduct startup optimisation. The shortcoming of this method is its low computation efficiency, i.e. many runs of simulation are needed to reach the optimal solution. SA is applied to the startup study on a pilot column with 20 bubble-cap trays for separating a methanol-water mixture (see Section 4.2).
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The reason we have used two different approaches to solve the startup optimisation problem is as follows. We have first implemented the startup model into a commercial software being able to carry out dynamic simulation. Thus the SA algorithm could be directly connected to the simulator. The computation time for the startup optimisation of one column (the bubble-cap tray column described in Section 4.2) was several days by a SUN SPARC 10 Station. It would be intolerable to use SA for the startup optimisation of the two-column system (described in Section 4.3). Therefore, we replaced SA with the gradient-based approach. Using this approach, the CPU-time for the startup optimisation of the two-column system was only several hours with the same workstation.
4. Experimental verification on different pilot plants In this section, experimental results on three different pilot plants are presented to verify the modelling and the optimisation approaches for startup of distillation columns. 4.1. A packed column The first pilot column is a packed column (height: 6.5 m, diameter: 0.07 m) which is packed with Sulzer packings (Sulzer Chemtech), 1.5 m BX in the rectifying section and 1.0 m DX in the stripping section. A total condenser is mounted inside the column top. The operating pressure is held with a vacuum pump. A frequency-adjustable divider for discharge (distillate) and return (reflux) of the condensate is employed to manipulate the reflux ratio. The reboiler is an electrical heat exchanger with adjustable heat duty. Several sampling and temperature measuring points are located along the column. The plant is equipped with a process control system. A mixture of two fatty alcohols (1hexanol (C6) and 1-octanol (C8)) is to be separated with the column at a specified steady state point. To startup the column from a cold state (e.g. 25 8C), it is reasonable that the maximum heat duty is to be used to warm up the packing, the column wall and the feed stream until the condensate reaches the top of the column. By using the maximum heat duty (/Q˙/ /1 kW) it takes about 30 min to receive the first drop of the condensate. Since the equilibrium model is used, the task of the optimisation is to determine the policies of the reflux ratio and the reboiler duty after this time point. The conventional direct setting startup procedure used in the chemical industry simply sets the values of the two control variables at the desired steady state (‘‘nominal’’ values). This procedure was tested on the column and it took 7 h to reach the desired steady state. Fig. 5 shows the measured temperature profiles along
Fig. 5. Measured temperature profiles of the packed column by direct setting policy.
the column resulted from this strategy. Kruse, Fieg & Wozny (1996) modified this procedure by first switching the heat duty to the nominal value but without reflux for a period of time, and then switching the reflux ratio to its nominal value. The time point of the second switching is determined by checking the summed minimum quadratic error of the measured and desired temperatures along the column. By implementing this strategy the startup period was reduced to 2 h. The sequential optimisation approach based on the equilibrium model is used to develop optimal startup policies. Corresponding to the height of the packings the number of the theoretical trays is 28 in the rectifying section and 22 in the stripping section. The influence of the packing on the holdup of the internal energy is also added to the model. The model is validated by comparing the data from simulation and experiment. For the dynamic optimisation, 40 time intervals are chosen to discretise the dynamic system. The nominal values corresponding to the steady state (R /1.5, Q˙/ /0.7 kW) are used as the guess profiles of the two control variables. The optimisation results are shown in Figs. 6 and 7 where t /0 corresponds to the time of the first drop of reaching the condenser. During the first 12 min the two
Fig. 6. Computed optimal reflux and reboiler duty policy for the packed column.
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Fig. 7. Computed optimal composition profiles of the packed column.
control variables should be kept at their limiting values (R /0.1, Q˙/ /1 kW) to run the column at the maximum speed. After this period the reflux ratio begins to increase and at the same time the reboiler duty begins to decrease, which slows down the run to prevent an overshooting. By doing so the column is allowed to slide to the desired steady state as fast as possible, because of the improvement of the separation effect. This can be clearly seen from Fig. 7: the distillate composition drops in the first 12 min due to the maximum heating and the minimum reflux, with which the bottom purity is increased quickly. In the second period the distillate composition returns to its specification due to the rising reflux ratio and the decreasing reboiler duty. From the computed results, the total startup period would be 52 min (i.e. 30 min to the first condensate plus 22 min to the steady state). The startup time of the implementation of the optimal policy to the pilot plant was found to be 1 h, as shown in Fig. 8. 4.2. A bubble-cap tray column The second pilot column we used for experimental verification is a tray-by-tray column. The column has a diameter of 100 mm and 20 bubble-cap trays with a
Fig. 8. Measured temperature profiles of the packed column by the optimal policy.
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central downcomer. Isolation coat is mounted to prevent the heat loss from the column wall. The boilup is provided by an electrical reboiler with a maximum duty of 30 kW. The condensation is carried out by a total condenser with cooling water. The plant is equipped with temperature, pressure, level and flow rate measurements and electrical valves for the flow control. All input/output signals are treated by a process control system. Several control loops have been configured and implemented on the plant. The control system is connected to the local area network to manage experimental data. The composition of the feed stream, distillate and bottom product is measured off-line with a gas chromatograph. We consider the separation of a mixture of water and methanol by this plant. The startup of the column to the steady state with a purity of 99.5 mol% for both methanol and water was studied. SA is used for the startup of this plant and the equilibrium model is used in the problem formulation. The model was implemented in the software SPEEDUP as a simulator, which is called by a file of the SA algorithm. The problem definition is the minimisation of the time period from an initial state of the column to the desired steady state. Fig. 9 shows the computed optimal operation policy and Fig. 10 shows the corresponding product purity profiles. It is interesting to note the difference between the optimal policies of the packed column (Fig. 6) and the tray column (Fig. 9). Unlike the operating policy for the packed column shown in Fig. 6, both the reflux rate and reboiler duty shown in Fig. 9 for this tray column should be high in the first period and should be decreased to the steady state nominal value in the second period. Figs. 10 and 11 shows the measured bottom and top temperature profiles by different startup policies: (a) direct setting; (b) zero reflux; (c) optimal policy. All three experiments had the same feed condition (composition: 29 mol% of methanol, flow rate: 15 l/ h, temperature: 60 8C). It can be seen that the time taken for reaching both bottom and top temperature at steady state was 220, 170 and 120 min by the three different policies, respectively, as shown in Fig. 12.
Fig. 9. Computed optimal policy for the bubble tray column.
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Fig. 10. Computed optimal purity profiles for the bubble tray column.
Fig. 13. The heat-integrated column system.
Fig. 11. Measured bottom temperature profiles of the bubble tray column.
of LP. Due to the heat-integration startup of the plant becomes complicated. The end of the startup period is defined as the time when both columns have arrived at the desired steady state. The startup time can be defined as the objective function subject to the steady state conditions. Since this kind of formulation may cause numerical expenses, the following formulation is used as an alternative to describe this optimisation problem: tf
g
2 2 2 LP HP min [(xHP D (xD x ) D (xB x ) B D x ) t0 2 (xLP B ]dt B x )
Fig. 12. Measured top temperature profiles of the bubble tray column.
4.3. A heat-integrated column system As shown in Fig. 13, the third pilot plant considered is a heat-integrated column system consisting of a high pressure (HP) and a low pressure (LP) column, with 28 and 20 bubble-cap trays, respectively. The vapour from HP is introduced as the heating medium to the reboiler
(15)
where xD and xB are the distillate and bottom product specifications, respectively, for the light component. t0 is the initial time point and tf is the final time point. The pilot plant with the parallel arrangement to separate a mixture of methanol and water is considered. The total feed flow is splitted into two parallel flows to the two columns and both columns have top and bottom product. This arrangement represents a typical operation case of such processes in the chemical industry. The input constraints are the limitations of the control variables, i.e. the reflux flow of both HP and LP and the reboiler duty of HP: 0 5LHP (t)5LHP max LP
0 5L (t)5LLP max 0 5QHP (t)5QHP max
(16) (17)
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The feed condition (flow, composition and temperature) is determined through experiment such that it will lead to a proper vapour and liquid load for the two columns. With the equilibrium model, we assume the starting state for the optimisation at the time point when the pressure of HP reaches 2.5 bar, at which both columns just approach VLE and the heat-integration takes place. The state variables at this initial state can be computed with the help of the measured data in the past experiment. Moreover, the profile of the pressure rising in the HP has to be estimated (Wendt et al., 2002). Furthermore, to deal with the time-optimal problem, a sufficiently large time horizon has to be chosen, in order to ensure the plant to reach the steady state. The final optimisation results of the optimal trajectories of the reboiler duty for HP and the reflux flow for both columns are shown in Fig. 14. The corresponding product composition and temperature profiles are shown in Figs. 15 and 16, respectively. As shown in Fig. 14, the optimisation results illustrate that at the beginning a high value of the reboiler duty should be chosen, since it is necessary to increase especially the bottom temperatures and thus the purity of the bottom products. With a slight time delay, the two refluxes need to be increased in order for the two columns to approach the desired purity of both top and bottom products as fast as possible, as shown in Figs. 15 and 16. However, as soon as the column system reaches close to the steady state, the controls need to be step by step decreased down to their steady state level. It should be noted that the optimisation results are developed based on the model. The model reflects the steady state points of the pilot plant fairly well and also depicts roughly the shape of the experimentally measured profiles, but there is still a large model-mismatch concerning the time delay. From the optimisation results, a startup rule for practical purpose can be obtained. It can be seen in Fig. 14 that there is one certain time point at which the control parameters have to be decreased drastically.
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Fig. 15. Computed optimal composition profiles during startup.
Fig. 16. Computed optimal temperature profiles during startup.
When this decrease for the reboiler duty has to be made, the temperature of the bottom in the LP has reached the value of approximately 98 8C. Since this temperature is the major concern for startup operation of this plant, this value can be used as a switching criterion. To transfer the optimisation results to a practical operation policy, an easy-implementing strategy can be derived, which means that the plant is first operated with the maximum value for all control parameters until the bottom of the LP reaches its switching temperature and then all the control parameters are switched to their steady state values.
Fig. 14. Computed optimal trajectories of the control variables.
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For comparison of startup time, different operating policies were implemented in different startup runs for the experimental verification. A feed flow of a methanol /water mixture (F /34.5 l/h, xf /0.3 mol/ mol, Tf /60 8C) was to be separated. The feed flows of the two columns are set as constant with their steady state value (F1 /19 l/h, F2 /15.5 l/h). The feed tray is set at 10th tray for HP and 6th tray for LP. The operating pressure at steady state is 4.7 bar for HP and atmospheric for LP. The startup of the pilot plant from a cold, empty state (at atmospheric temperature and pressure) to the desired steady state (xD /0.99, xB / 0.01 mol/mol for both columns) was considered. Based on simulation as well as experimental results, the bottom and top temperatures of the two columns corresponding to the steady state purity specifications are THP Bss / LP LP 144 8C, THP Tss /108 8C and TBss /99 8C, TTss /65 8C, respectively. And the corresponding values of the control variables at this steady state are QHP ss /9.54 LP kW, LHP ss /26 l/h, Lss /11.2 l/h. During startup, the level and flow control loops shown in Fig. 13 are active, while the temperature control loops (except for the cooling water temperature control) are set to open loop in order to manually implement the startup policies. The direct setting strategy was first implemented for startup. Fig. 17 shows the measured temperature profiles of the two columns. It can be seen that there is a time delay of about 50 min for the top temperature of LP (TLP T ) to begin to increase after the top temperature of HP (THP T ) rises. This is because the driving force of LP is from the latent heat of the vapour from HP. first reached its However, despite of the delay, TLP T steady state value. It took 465 min for all temperatures to approach their steady state. Moreover, it is shown that the warm-up time of the two columns (i.e. the
discontinuous phase) was about 10 and 20 min, respectively. The second strategy studied was the total distillate strategy. In a beginning period the condensed liquid from both columns was pulled out as distillate (zero reflux) and then the operation was switched to the reflux flow for the steady state value. The reboiler duty remained constant during startup. This total distillate strategy has the advantage to accelerate the rise of the bottom temperature of both columns. The switching point from zero reflux to the nominal reflux value was decided by the minimum point of the following function: MT
N X
½Ti Tss;i ½
(19)
i 1
which is the sum of discrepancies of measured temperatures on all trays (there is a temperature sensor on each tray for the pilot plant) and their desired values at the steady state. The value of this function was on-line computed and observed, through which the switching time point can be determined. Fig. 18 shows the measured temperature profiles caused by the total distillate strategy. Due to zero reflux before the switching, the temperature increased with a fast speed. An over-shooting of the top temperature of both columns was observed. Because of the heat-integration, the bottom temperature of LP is stagnated and thus without over-shooting. Switching the reflux flow to their steady state value led to a decrease of the top temperature of both columns. The total startup time for all temperatures to reach their steady state value was reduced to 403 min. The optimal strategy developed by the optimisation was implemented in the third experimental run. From the optimisation results shown in Fig. 14, the control
Fig. 17. Measured temperature profiles by direct setting strategy.
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263
Fig. 18. Measured temperature profiles by total distillate strategy.
variables (the reflux flow for both columns and the reboiler duty for HP) should be first set at an optimised maximal value for a time period and then decreased gradually to their steady state value. From the practical point of view, it is desired to implement a simple startup policy. Therefore, we simply tailored the numerically optimised operating policy shown in Fig. 14 into a twostage strategy. The optimised maximal values of the HP control variables (QHP opt /11.14 kW, Lopt /29 l/h, LP Lopt /13.5 l/h) were taken in the first period and the steady state values were set in the second period to the pilot plant. The switching time point was chosen at the time when the bottom temperature of LP reaches 98 8C that is almost approaching its steady state value.
Fig. 19 shows the top and bottom temperature profiles of both columns by the optimal strategy. With HP the enhanced values of the control variables, THP B , TT LP and TB rose fast to their steady state value before the switching. After the switching the two columns continued running to the desired steady state. It can be seen that HP ran a little bit over the steady state before the switching in order to provide enough energy to LP, so that both columns could approach the desired steady state as quickly as possible. Compared with the results of the other startup strategies (Figs. 17 and 18), the optimal strategy really resulted in the best startup performance and thus took the shortest startup time. With the optimal strategy, the total startup time was
Fig. 19. Measured temperature profiles by optimal strategy.
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about 300 min, which is 64% of the time needed by the direct setting strategy.
the desired steady state. Moreover, the startup of some distillation columns with special characteristics such as reactive distillation and three-phase distillation is also a future challenge.
5. Conclusions The theoretical work on dynamic process modelling, simulation and optimisation has been well-developed in the last two decades. However, very few studies on the validation and implementation of such results to real plants for column startup have been made in the past. Although there exist profit incentives, no industrial application has been reported on realisation of dynamic optimisation results. Extensive and intensive work on practical verification needs to be conducted to bridge the theoretical development and the industrial application, so as to convince the process industry to apply the theoretical results. To this end, beside the theoretical work of modelling, simulation and optimisation, practical issues including plant and equipment engineering should be considered. In this work, we have considered the startup optimisation problem, which represents one of the most complicated dynamic operations in chemical industry. Dozens of hours are needed in industrial practice for column startup and it results in large amount of off-spec products. Many factors have impacts on the performance of startup operations. There exist no general rules of startup strategies for all kinds of distillation columns. We used a systematic approach including modelling, model validation, optimisation and implementation on the real pants to address this problem. Model-based optimisation for searching time-optimal policies for column startup was the core of this work. A detailed equilibrium model was chosen for the base of optimisation. The model was at first validated (e.g. tray efficiency, tray holdup, column heat loss etc.) and then used for startup policy development. Reflux and reboiler duty policies for startup were searched for and verified on different pilot plants. The temperature profiles on trays of the column were measured as signals to observe the performances by different startup strategies. Significant reduction of startup time was achieved by implementing the optimal policies, in comparison to the conventional startup strategy. The experimental results demonstrated the applicability of the modelling and optimisation results, showing that they can be applied to start up industrial columns. An extension of this work will be the optimisation of column startup based on the non-equilibrium model, so as to include the time period from the cold, empty state to the equilibrium state into the problem formulation. Moreover, heuristics for column startup is to be developed. These heuristics may depend on the properties of equipment (packings, tray-internals), the features of mixtures to be separated, as well as the requirement of
Acknowledgements We thank Deutsche Forschungsgemeinschaft (DFG) for the financial support in this work under the contract WO565/6-3 and WO565/10-3.
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